NLC2-Decomposition in Polynomial Time

NLCk is a family of algebras on vertex-labeled graphs introduced by Wanke. An NLC-decomposition of a graph is a derivation of this graph from single vertices using the operations in question. The width of the decomposition is the number of labels used, and the NLC-width of the graph is the smallest width among its NLC-decompositions. Many difficult graph problems can be solved efficiently with dynamic programming if an NLC-decomposition of low width is given for the input graph. It is unknown though whether arbitrary graphs of NLC-width at most k can be decomposed with k labels in polynomial time. So far this has been possible only for k = 1, which corresponds to cographs. In this paper, an algorithm is presented that works for k = 2. It runs in O(n4 log n) time and uses O(n2) space. Related concepts: clique-decomposition, clique-